Lemma 93.13.2. Let $k$ be a field. Let $B$ be a finite type $k$-algebra. Let $S \subset B$ be a multiplicative subset ideal such that if $\mathop{\mathrm{Spec}}(B) \to \mathop{\mathrm{Spec}}(k)$ is not smooth at $\mathfrak q$ then $S \cap \mathfrak q = \emptyset $. Let $N$ be a finite $B$-module. Then there is a canonical bijection
Proof. This proof is the same as the proof of Lemma 93.12.4 but easier. We suggest the reader to skip the proof. The map is given by localization: given $0 \to N \to C \to B \to 0$ in $\text{Exal}_ k(B, N)$ we send it to the localization $S_ C^{-1}C$ of $C$ with respect to the inverse image $S_ C \subset C$ of $S$. Compare with the proof of Lemma 93.8.7.
The smooth locus of a morphism of schemes is open by definition. Let $J \subset B$ be an ideal cutting out the set of points in $\mathop{\mathrm{Spec}}(B)$ where $\mathop{\mathrm{Spec}}(B) \to \mathop{\mathrm{Spec}}(A)$ is not smooth. Since $k \to B$ is of finite presentation the complex $\mathop{N\! L}\nolimits _{B/k}$ can be represented by a complex $N^{-1} \to N^0$ where $N^ i$ is a finite $B$-module, see Algebra, Section 10.134 and in particular Algebra, Lemma 10.134.2. As $B$ is Noetherian, this means that $\mathop{N\! L}\nolimits _{B/k}$ is pseudo-coherent. For $g \in J$ the $k$-algebra $B_ g$ is smooth and hence $(\mathop{N\! L}\nolimits _{B/k})_ g = \mathop{N\! L}\nolimits _{B_ g/k}$ is quasi-isomorphic to a finite projective $B$-module sitting in degree $0$. Thus $\text{Ext}^ i_ B(\mathop{N\! L}\nolimits _{B/k}, N)_ g = 0$ for $i \geq 1$ and any $B$-module $N$. Finally, we have
The first equality by More on Algebra, Lemma 15.99.2 and Algebra, Lemma 10.134.13. The second because $\text{Ext}^1_ B(\mathop{N\! L}\nolimits _{B/k}, N)$ is $J$-power torsion and elements of $S$ act invertibly on $J$-power torsion modules. This concludes the proof by the description of $\text{Exal}_ A(B, N)$ as $\text{Ext}^1_ B(\mathop{N\! L}\nolimits _{B/A}, N)$ given just above Lemma 93.12.4. $\square$
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